| L(s) = 1 | + (−153. − 266. i)2-s + (−3.10e4 + 5.37e4i)4-s + (8.07e4 − 1.39e5i)5-s + (−1.06e6 − 1.84e6i)7-s + 9.01e6·8-s − 4.97e7·10-s + (5.37e7 + 9.30e7i)11-s + (−5.53e6 + 9.59e6i)13-s + (−3.27e8 + 5.66e8i)14-s + (−3.71e8 − 6.44e8i)16-s + 1.69e9·17-s + 2.66e9·19-s + (5.00e9 + 8.67e9i)20-s + (1.65e10 − 2.86e10i)22-s + (1.38e10 − 2.39e10i)23-s + ⋯ |
| L(s) = 1 | + (−0.850 − 1.47i)2-s + (−0.946 + 1.64i)4-s + (0.462 − 0.800i)5-s + (−0.487 − 0.844i)7-s + 1.52·8-s − 1.57·10-s + (0.831 + 1.43i)11-s + (−0.0244 + 0.0424i)13-s + (−0.829 + 1.43i)14-s + (−0.346 − 0.600i)16-s + 1.00·17-s + 0.683·19-s + (0.875 + 1.51i)20-s + (1.41 − 2.44i)22-s + (0.848 − 1.46i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.358445913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358445913\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (153. + 266. i)T + (-1.63e4 + 2.83e4i)T^{2} \) |
| 5 | \( 1 + (-8.07e4 + 1.39e5i)T + (-1.52e10 - 2.64e10i)T^{2} \) |
| 7 | \( 1 + (1.06e6 + 1.84e6i)T + (-2.37e12 + 4.11e12i)T^{2} \) |
| 11 | \( 1 + (-5.37e7 - 9.30e7i)T + (-2.08e15 + 3.61e15i)T^{2} \) |
| 13 | \( 1 + (5.53e6 - 9.59e6i)T + (-2.55e16 - 4.43e16i)T^{2} \) |
| 17 | \( 1 - 1.69e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 2.66e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + (-1.38e10 + 2.39e10i)T + (-1.33e20 - 2.30e20i)T^{2} \) |
| 29 | \( 1 + (-8.55e10 - 1.48e11i)T + (-4.31e21 + 7.47e21i)T^{2} \) |
| 31 | \( 1 + (8.62e10 - 1.49e11i)T + (-1.17e22 - 2.03e22i)T^{2} \) |
| 37 | \( 1 - 7.44e10T + 3.33e23T^{2} \) |
| 41 | \( 1 + (-4.72e11 + 8.17e11i)T + (-7.77e23 - 1.34e24i)T^{2} \) |
| 43 | \( 1 + (-1.38e11 - 2.39e11i)T + (-1.58e24 + 2.75e24i)T^{2} \) |
| 47 | \( 1 + (-1.76e11 - 3.06e11i)T + (-6.03e24 + 1.04e25i)T^{2} \) |
| 53 | \( 1 + 5.82e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + (1.96e12 - 3.40e12i)T + (-1.82e26 - 3.16e26i)T^{2} \) |
| 61 | \( 1 + (7.92e12 + 1.37e13i)T + (-3.01e26 + 5.21e26i)T^{2} \) |
| 67 | \( 1 + (-2.29e13 + 3.98e13i)T + (-1.23e27 - 2.13e27i)T^{2} \) |
| 71 | \( 1 + 3.72e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.00e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + (-8.56e13 - 1.48e14i)T + (-1.45e28 + 2.52e28i)T^{2} \) |
| 83 | \( 1 + (3.77e13 + 6.53e13i)T + (-3.05e28 + 5.29e28i)T^{2} \) |
| 89 | \( 1 - 2.19e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + (4.56e14 + 7.90e14i)T + (-3.16e29 + 5.48e29i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.75936768432219195855926862234, −12.27745366258102115706672270508, −10.62969576012911465113114620500, −9.723753966715507908228839819179, −8.850736439590673760179056650464, −7.10394175601364691911436797770, −4.68917975436008704568814422095, −3.25381526274607456968485866637, −1.59101614088758018697178164490, −0.815436974562007768910338152662,
0.868936151560604166200901179421, 3.07893793270901555931451890249, 5.70285473919582150648845215365, 6.26745404919303803309595797005, 7.66940510849985536330397676531, 9.003460567144470997822076557613, 9.881207407853518584720139573590, 11.57839727083133730206504792644, 13.68439473255758070550801542401, 14.61753947506514970752236486175
